Presentation on theme: "Instructor: Dr Sachin S. Talathi"— Presentation transcript:
1Instructor: Dr Sachin S. Talathi BME 6938 NeurodynamicsInstructor: Dr Sachin S. Talathi
2Recap Neurons are excitable cells Neuronal classification Communication in the brain is mediated by synapses: Electrical and Chemical
3Neuronal SignalingNeuronal signaling is mediated by the flow of dissociated ions across the cell membrane.The fundamental laws that govern the flow of these ions through the cell membrane are:Ficks Law of DiffusionOhms Law of DriftSpace charge neutralityEinsteins Relation between diffusion and drift
4Fick’s Law of Diffusion Fick’s law relates the diffusion gradient of ions to their concentration.Jdiff: Diffusion flux, measuring the amount of substance flowing across unit area per unit time (molecules/cm2s )D: Diffusion coefficient ( cm2/s )[C]: Concentration of the ions (molecules/cm3 )The negative sign indicates the flow of ions across decreasing concentration gradientFicks Law Animation
5Ohms law of drift (Microscopic view) Charged particle in the presence of external electrical field E experience a force resulting in their drift along the E field gradientJdrift: Drift flux, measuring the amount of substance flowing across unit area per unit time ( molecules/cm2s )mu: electrical mobility of charged particle(cm2/sV)[C]: Concentration of the substance (ions) (molecules/cm3 )z: Valence of ionThe negative sign indicates the flow of ions across decreasing voltage gradient. The ion flux depends on electrical resistance which in turn depends on ion mobility
6Space charge neutrality Biological systems are overall electrically neutral; i.e., the total charge of cations in a given volume of biological material equals the total charge of anions in the same volume biological material
7Einstein relationIt relates the diffusion constant (effect of motion due to concentration gradients) of an ion to its mobility (effect of motion due to electrical forces) Basic idea is that the frictional resistance created by the medium is same for ions in motion due to drift and diffusion.
8Some high-school chemistry 1 mole= Avogadro’s number (NA) of basic units (atoms, molecules, ions…)Concentration is typically given in units of molar.1 Molar=1 mole/litre=10-3 moles/cm3Relation between gas constant (R) and Boltzmann’s constant (k): R=kNAFaraday constant F: Magnitude of one mole of charged particles: F=qNA
9Some-algebraMembrane capacitance of a cell membrane is around 1 microF. Concentration of ions within and outside of a cell is 0.5 M. Determine the fraction of free (uncompensated) ions required on each side of a spherical cell of radius 25 micro m to produce 100mV? Ans: ~ % For realistic cell dimension, from above calculations we see that generation of 10s of mV of voltage does not violate space-charge neutrality (~99.9% of charges are compensated)
10Nernst-Plank Equation Nernst Plank equation governs the current generated from the flow of individual ions across the cell membrane. Continuity Equation The time dependent Nernst Plank Equation:Derive the eq in class
11Nernst EquationNernst equation is special case of NPE, where in the membrane potential is obtained as a function of concentration gradient across the cell membrane when the net current flow generated by the ion is zero.
12Typical scale of reversal potential values Question:What is the direction of flow of following ions under normal conditions?1.Na+2. K+3. Ca2+4. Cl-(Hint: Look at the chart of reversal potentials and Nernst Equilibrium potential equation)At 37 oCThe negative sign indicates the flow of ions across decreasing concentration gradientmV
13Specific Example Ion concentration for cat motoneuron: Vm=-70 mV Nernst Potential: At body temperature 37oCInside mol/m3Outside mol/m3Na+15150K+5.5Cl-9125Specific problem: The resting potential is not equal to reversal potential for Na and K; what does that mean??How is resting potential achieved?
14Ion distribution and Gradient maintenance Active Transport:Flow of ions against concentration gradient.Requires some form of energy sourceExamples: Na+ pumpRead section in Johnston’s& Wu book for more information.Passive Transport:Selective permeability to some ions results in concentration gradientNo energy source requiredPassive distribution of ions can be determined using the Donnan rule of equilibriumIon pumps are proteins inside the plasma membrane of cells, which pump ions across the plasma membrane often against the concentration gradient. Read sectio
15Donnan Equilibrium Rule The membrane potential equals the reversal potential of all ions that can passively permeate through the cell membrane.Mathematically the Donnan Rule implies:Have a look at Donnan Rule in works; through animaltion developed by Larry Keeley:
16Graphical illustration of ion gradient maintenance
17Example: Application of Donnan Rule Consider a two compartment system separated by a membrane that is permeable to K+ and Cl- but is not permeable to a large anion A-. The initial concentrations on either side of membrane are:Is the system in electrochemical equilibrium (no ion flow across the membrane?If not, what direction the ions flow? And what are the final equilibrium concentrations?Ion typeI (conc in mM)II (conc in mM)A-100K+150Cl-50
18Steady state solution to NPE We can solve the NPE equation in steady state (ie no time dependence for concentrations). A special case was seen through NE, wherein in addition to the membrane being in steady state, the membrane was in equilibrium (I=0).
19Boundary conditions-Cell membrane Permeability P: Defined as absolute magnitude of ion flux when there is a unit concentration difference between internal and external fluids and the concentration is linear function of distance within the membrane(in molar units): Partition coefficient, which measures the drop in concentration of species at fluid membrane interface.
20Constant field assumption Most common assumption is charge density within the membrane is identically zero. This assumption leads to the following expression for V(x) with boundary conditions: V(x=0)=Vm and V(x=l)=0We get the following expression for current:whereNote: Poisson equation from Electrostatics:
21Goldman Hodgkin Katz Model It is widely used model to predict the resting membrane potential Vm for nerve cellsThe formula for Vm is derived as a solution NPE with constant field assumptionFurther more it is assumed that ions flow across the cell membrane without interacting with each otherFor membrane that is permeable to M positive monovalent ions and N negative monovalent ions, the GHK formula for Vm in equilibrium conditions is:Derive this equation in class
22Exercise (extra credit) Show that the resting equilibrium membrane potential Vm for a cell membrane that is permeable to monovalent and divalent ions is given by
23Application of the GHK equation Lets use GHK eqn to determine the contribution to membrane potential from active ion transport mechanism’s.Na-pump result in flow of 3 Na+ ions across the cell membrane for every 2K+ ions. What is the resulting equilibrium potential of the cell of squid axon for which the concentration gradients across the cell are:The permeability ratio is Pk:Pna=1:.03Ion typeInside (mM)Outside (mM)K+40020Na+50550